Design of fuzzy PD-controlled overhead crane system with anti-swing compensation

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1 Journal of Engineering and Computer Innovations Vol. (3), pp , March 11 Available online at ISSN Academic Journals Full Length Research Paper Design of fuzzy PD-controlled overhead crane system with anti-swing compensation Shebel Asad*, Maazouz Salahat, Mohammad Abu Zalata, Mohammad Alia and Ayman Al Rawashdeh Department of Mechatronics, Faculty of Engineering Technology (FET), Al Balqa Applied University (BAU), P. O Bo 158. Amman - Jordan. Accepted 4 February, 11 This work aimed to find a proper control strategy to transfer loads using overhead cranes. The proposed control strategy, which is Fuzzy PD-based, should take into account two main factors. First, the time needed to move the payload from the initial pick up point to the destination point that must be minimized. Secondly, the oscillation of the payload must be reduced to prevent hazards for people and equipment in the work place. The current work, presents a comparative analysis of fuzzy PD based control basing on classical PD approach. A simplified model has been derived. The proposed control techniques have been designed and validated with MatLab. Numerical comparative results have been obtained and discussed. Key words: Crane, position control, anti-swing compensation, fuzzy PD- based control. INTRODUCTION One of the current trends in industry is that cranes became larger and higher. So they supposed to be faster in order to achieve acceptable transfer times. Unfortunately, cranes with large structures that are moving at high speeds are associated with undesirable payload oscillations resulting from the system dynamics. Cranes can be classified in terms of their mechanical structures and dynamics into three types: Overhead, rotary, and boom cranes. Overhead crane (OHC) is a common type of cranes used to transfer the payload from one position to desired position. A gantry crane incorporates a trolley which moves along the track and translates in a horizontal plane. In such cranes system, the load suspended from the trolley by cable is subject to swing caused by improper control input and disturbances. The failure of controlling crane may cause accident and may harm people and the surroundings. Therefore, such crane control must be able to move the trolley adequately fast and to suppress the payload swing at the final position. This is so-called anti- *Corresponding author. shebel_asad@hotmail.com. swing control. In the present work, a Fuzzy PD (Proportional- Derivative) has been designed basing on classical PD approaches to ensure a robust and smooth position control of the carte (trolley position) with the presence of another fuzzy-based anti-swing compensator. The proposed model has been validated with MatLab/Simulink, fuzzy logic and virtual reality toolboes Numerical comparative results have been obtained to be found very accepted. To understand the overhead crane system, a simplified dynamic model has been derived that encounters the reality and complicities might be found with such type of cranes. The main feature of the derived model is its capability to match real system s nonlinearity (trolley position and swinging angle). DERIVATION OF A SIMPLIFIED DYNAMIC MODEL FOR OHC The first step in deriving the equations of motion using the Lagrangian approach is to find the positions of the

2 5 J. Eng. Comput. Innov. With this assumption, the simplified model of motion for the overhead crane system can be obtained as in Equation 4: ( Trolley θ ( Swing position angle ) : ) : ( M + m ) && + ml && θ = F && + L && θ + gθ = (4) Figure 1. Presents the configuration of the approimated model of OHC. load and position of the trolley with respect to the reference point, then the kinetic and potential energies and dissipation function of the whole system would be found. Finally, the equations of motion are as given in 1 and (Moustafa et al., 9; Abdel-Rahman et al., 3). && θ = && = F F cos( θ) + ( M + m) gsin( θ) + ml& θ L( M+ msin ( θ)) L ( M + m sin Equations (1) and () represent a nonlinear mathematical model of the overhead crane. Where; F is the applied mechanical force in Newton, θ is the swinging angel of the payload in radians, m is the payload mass, M is the trolley mass, is the trolley position and L is the robe length in meters. Approimation of the full model sin( θ)cos( θ) + mg sin( θ ) cos( θ ) + ml & θ sin( ( θ )) θ ) The aforementioned derived model given in Equations 1 and is an etremely nonlinear dynamic model. The nonlinear model could be approimated to simplify the progress of modeling in order to be used for numericalcontrol purposes (Kiviluoto, 9). Before the model linearization, a hypothesis for safe and smooth operation will be assumed and the swing angle would be kept small. As given in set of Equation (3). sin( cos( θ& θ ) θ ) θ 1 (1) () (3) State space representation of the system After the simplification, Equation 4 can be re-written in state space representation as in Equation 5, and the equation of motion could be then obtained as shown in Equation 6: 1 Trolley position & = = Trolley velocity = θ 3 Swing angle 4 θ & Rate _ Swing angle & & 1 & && & = = = & 3 θ& & && 4 θ 1 m g M M + m g M L 1 1 M M L Output equation (y = C+Du) is as given in Equation 7, y = 1 1 & θ θ& From the previously described equations, it can be understood that the load (swing) angle (in other words, vibration that could be occurred) depends on the applied force F, the trolley and load's mass, the gravitational force and the length of the rope. The final simplified model can be built as shown in Figure 1, With a model, shown in Figure 1, the trolley position would move to infinity, while the load swinging is within -.7 to.7 radians. So, since the trolley is non-stop at any location, it seems to be necessary if the input to be modified in order to make the trolley stop, or in the worse condition, the trolley needs to be stopped at somewhere. This will lead naturally to select a rectangular or bangbang forcing function as an input vector. It seems that there is no mechanism or factor that makes the trolley F (5) (6) (7)

3 Asad et al 53 Figure. New system configuration. stop (because the system model took an ideal situation). Regarding on this, this input has been introduced in order to cope with this condition. In other point of view, this type of input is well known as a time-optimal solution (Fang et al., 3; Uchiyama, 9). Regarding on this matter, modification of the selection of input shape has been done. This involves changing in system s block diagram, as well as the input itself only. Figure shows the new Simulink block diagram and the input force applied to the system. hoisting rope decreased from.3 to. m. However the fluctuation of the trolley increased with shorter rope. This is due to the load oscillation increased when the rope is shorter. As shown from Figure 3d when the load mass increased from.8 to 1.5 kg which means that the load mass is greater than the trolley mass the trolley (m < M) cannot reach more than.4 m and vibrate with obvious fluctuation, and the swing angle has high magnitude (.9 rad) and frequency after the input force taken off. Validation of the simplified model The effect in variation of input force, length of the hoisting rope, trolley mass and load mass on the dynamic behavior of the overhead crane are investigated. The comparative results are as shown in Figure 3. Figure 3 shows the response of the system in term of trolley position and swing angle when different input forces applied to the system (., 1 and 5 N). We notice that when the input force is increase from 1 to 5 N, the trolley is able to travel more distance where it can reaches around.8 m from its initial position. However the vibration of the trolley is greater. The magnitude of the load swing also increased to around.8 rad. But the frequency of the load remains the same. When input force decreased from 1 to. N the trolley will only be able to reach around.1 m from its initial position. The magnitude of the load swing is decreased to approimate.33 rad. However the frequency of the load swing remained the same. From Figure 3c, we can see that the magnitude (. rad) and frequency of the swing angle is greater. Trolley still can reach at.58 m at s although the length of the CONTROL DESIGN Proposed control structure The structure of the proposed controller for the overhead crane system is shown in Figure 4. The proposed controller consists of fuzzy logic controllers for both position and anti-swing control respectively. The objective of the proposed fuzzy logic controllers is to control the payload position X(s) so that it moves to the desired position X ref(s) as fast as possible without ecessive swing angle of the payload θ(s). Here, the design of fuzzy logic control is based on a classical control approach (proportional derivative control mode PD). It shows that fuzzy logic controller is a controller that may realize the skill of human operators and the design rules describe the subjective fuzziness of operators eperiences instead of the use of mathematical model of the plant as modern control theory approaches. Fuzzy PD controllers Fuzzy PD controllers are physically related to classical PD controller. The parameters settings of classical and fuzzy controllers are based on deep common physical background. The introduced method considerably simplifies the setting and realization of fuzzy PD controllers (Pivoňka, 1999, 1998; Pivoňka et al., 1998). A classical ideal PD controller is described as noted in Equation 8.

4 54 J. Eng. Comput. Innov. Time (s) (a) Time (s) (c) Time (s) Time (s) Time (s) Time (s) Time (s) (b) (d) Figure 3. (a) Bang-bang with different input forces 1, 5 and. N. (b) Trolley position and swinging angle at different F. (c) Trolley position and swinging angle with different rope length (L=.,.3 and.5 m). (d) Trolley position and swinging angle with different load masses (m=.3,.8 and 1.5 kg).

5 Asad et al 55 Figure 4. Proposed fuzzy PD-based overhead control system. Figure 5. Fuzzy PD controller structure with the normalized universe range. u(t) = K (e(t) + T D.e (t)) We would like to know when the action value is equal to zero, that is why we can write: e(t) + T D.e (t) = The solution of Equation (9) is e (t) = e(t)/t D (8) (9) (1) The Equation (1) depends only on the derivation time constant of the PD controller and its physical meaning is similar to Equation (4) for the PI controller. If we transfer the Equation (8) to the discrete form, we get an equation of the discrete PD controller u(k) = K (e(k) + T D e(k)) (11) Where e(k) = (e(k) - e(k-1))/t and T is the sample period. In the net step we map the rule base to the discrete state space e(k), e(k). We initiate the scale M for the universe range, M >. This scale sets ranges for the error and the change-in-error. After etending the Equation (11) we get u(k) = KM (e(k)/m + T D. e(k)/m) (1) We apply fuzzification to input variables and after defuzzification we get the equation u(k) = KM D{F{e(k)/M +T D e(k)/m}} (13) The realization of Equation 13 of the fuzzy PD controller is depicted in Figure 5. The adjustment of fuzzy controllers may be considerably simplified when fuzzy controller with a unified universe is used. The parameters to be tune then have their physical meaning and fuzzy

6 56 J. Eng. Comput. Innov. (a) (b) Figure 6. (a) Fuzzy sets for the proposed fuzzy PD-based position control (b) and either anti-swing compensator. controller can be approimately adjusted using known rules for classical PD controllers. Suitable choice of inference method can ensure behavior, which is close to one of classical PD controller for both the tracking problem and the step disturbance rejection (Pivoňka, 1998; Pivoňka et al., 1998). The fuzzy sets are, as shown in Figure 6, assumed to have initially symmetrical layout and Ziegler - Nichols method, tunes the parameters of the proposed fuzzy PD-based position controller and either anti-swing compensator. SIMULATION RESULTS The aforementioned model concerning the fuzzy PDbased overhead crane system has been designed and validated with MatLab/Simulink. The obtained comparative results for both fuzzy PD and classical PD, given in Figure 7, have been etracted with considering the tracking problem and the step disturbance rejection.

7 Asad et al 57 (a) Controlled position (b) Swinging angle Figure 7. Simulation comparative results of the proposed Fuzzy PD and classical PD at set point tracking (X ref) and step disturbance effect (as shown inside the defined ellipse). As concluded from Figure 7a, it could be seen that the performance of the closed loop control system using Fuzzy PD, when tracking the reference position X ref, is being enhanced with modification in dynamic state comparing with classical PD control structure. While the effect of the swing angle has been well eliminated using the proposed Fuzzy PD-controlled structure comparing with classical PD one (Figure 7b). COMPARATIVE ANALYSIS Table 1 summarizes the net comparative results shown previously in Figure 7. Where a comparative study for conventional PD controller with fuzzy PD one had been affected, then the comparative results are found at different set point values (X ref ). When considering: rising time in seconds, maimum overshoot, maimum overshooting in case of disturbance submission and steady-state error, it could be conclude that the overall steady state and dynamic performance of the proposed Fuzzy PD controlled plant seems to be better than that it was with classical approach (PD). These results that are being summarized in Table 1 have been validated at different carte positions (for eample, 5, 5 and 5).

8 58 J. Eng. Comput. Innov. Table 1. Comparative analysis for fuzzy PD and classical PD-based overhead crane system. X ref Controller Rising time (s) Maimum over shoot (m) Maimum over shoot for the disturbance (m) Steadystate error ss 5 PD Fuzzy PD ss<.1 5 PD Fuzzy PD PD Fuzzy PD CONCLUSION It could be noticed the fuzzy PD-based control approach has enhanced the robustness and reduced the settling time. The sensitivity of the closed-loop response against an effectiveness parameter variation and disturbances has been also modified. The used method with the unified universe of discourse, stated in this article, considerably simplifies setting of fuzzy PD controllers. It allows approimate adjustment of controller's parameters according to wellknown methods for PD controller synthesis. If the universe has non-linear membership function layout then the results can have better behaviour than the classical PD controller. The fuzzy PD controller can be programmed like a unified block in a controller and therefore work consumed on an implementation to the particular control system. Abdel-Rahman EM, Nayfeh AH, Masoud ZN (3). Dynamics and control of cranes: Rev. J. Vib. Control, 9(7): Fang Y, Dion WE, Dawson DM, Zergeroglu E (3). Nonlinear coupling control laws for an underactuated overhead crane system. IEEE/ASME Trans. Mechatron., 8(3): Kiviluoto S (9). Modelling and control of vertical oscillation in overhead cranes. M.Sc. Thesis. Helsinki University of Technology, Espoo, Finland, pp Uchiyama N (9). Robust control for overhead cranes by partial state feedback. Proceedings of the Institution of Mechanical Engineers, Part 1. J. Syst. Control Eng., 3(4): Pivoňka P (1999). Physical Background of Fuzzy PI and PD Controller. IFSA Eighth International Fuzzy Systems Association World Congress. Taipei, Taiwan, pp Pivoňka P (1998). Fuzzy PI+D controller with a normalized universe. European Congress EUFIT'98. Aachen, Germany, : Pivoňka P, Findura M (1998). The alternative realization of fuzzy controllers. Automatizace. Czech Republic, 41(1,11,1): REFERENCES Moustafa AF, Trabia B, Ismail IS (9). Modelling and control of an overhead crane with a variable length fleible cable. Int. J. Comput. Appl. Technol., 34(3): 16-8.